Solve ∫ (from 0 to 2 pi) of (cost)(sint)^4]1dt

$$\int _ { 0 } ^ { 2 \pi } ( \cos t ) ( \sin t ) ^ { 4 } ] 1 d t$$

Evaluate

$$\pi \sin(2t)\left(\sin(t)\right)^{3}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \cos(t)\left(\sin(t)\right)^{4}\mathrm{d}x$$

Find the integral of \cos(t)\left(\sin(t)\right)^{4} using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\cos(t)\left(\sin(t)\right)^{4}x$$

The definite integral is the antiderivative of the expression evaluated at the upper limit of integration minus the antiderivative evaluated at the lower limit of integration.

$$2\cos(t)\left(\sin(t)\right)^{4}\pi +0\cos(t)\left(\sin(t)\right)^{4}$$

Simplify.

$$\pi \sin(2t)\left(\sin(t)\right)^{3}$$

Differentiate w.r.t. t

$$\pi \left(\sin(t)\right)^{2}\left(2\sin(t)\cos(2t)+3\sin(2t)\cos(t)\right)$$